%I A000125 M1100 N0419
%S A000125 1,2,4,8,15,26,42,64,93,130,176,232,299,378,470,576,697,834,988,1160,
%T A000125 1351,1562,1794,2048,2325,2626,2952,3304,3683,4090,4526,4992,5489,6018,
%U A000125 6580,7176,7807,8474,9178,9920,10701,11522,12384,13288,14235,15226
%N A000125 Cake numbers: maximal number of pieces resulting from n planar cuts through 
               a cube (or cake): C(n+1,3)+n+1.
%C A000125 Note that a(n) = a(n-1) + A000124(n-1). This has the following geometrical 
               interpretation: Define a number of planes in space to be in general 
               arrangement when (1) no two planes are parallel, (2) there are no 
               two parallel intersection lines, (3) there is no point common to 
               four or more planes. Suppose there are already n-1 planes in general 
               arrangement, thus defining the maximal number of regions in space 
               obtainable by n-1 planes and now one more plane is added in general 
               arrangement. Then it will cut each of the n-1 planes and acquire 
               intersection lines which are in general arrangement. (See the comments 
               on A000124 for general arrangement with lines.) These lines on the 
               new plane define the maximal number of regions in 2-space definable 
               by n-1 straight lines, hence this is A000124(n-1). Each of this regions 
               acts as a dividing wall, thereby creating as many new regions in 
               addition to the a(n-1) regions already there, hence a(n)=a(n-1)+A000124(n-1). 
               - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
%C A000125 More generally, we have: A000027(n) = binomial(n,0) + binomial(n,1) (the 
               natural numbers), A000124(n) = binomial(n,0) + binomial(n,1) + binomial(n,
               2) (the Lazy Caterer's sequence), a(n) = binomial(n,0) + binomial(n,
               1) + binomial(n,2) + binomial(n,3) (Cake Numbers). - Peter C. Heinig 
               (algorithms(AT)gmx.de), Oct 19 2006
%C A000125 If Y is a 2-subset of an n-set X then, for n>=3, a(n-3) is the number 
               of 3-subsets of X which have no exactly one element in common with 
               Y. - Milan R. Janjic (agnus(AT)blic.net), Dec 28 2007
%C A000125 a(n) is the number of compositions (ordered partiitions) of n+1 into 
               four or fewer parts or equivalently the sum of the first four terms 
               in the nth row of Pascal's triangle. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Jan 23 2009]
%C A000125 {a(k): 0 <= k < 4} = divisors of 8. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 17 2009]
%D A000125 R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventues in 
               Applied Mathematics, Princeton Univ. Press, 1999. See p. 27.
%D A000125 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%D A000125 H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 
               177.
%D A000125 D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 
               292-298.
%D A000125 S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 
               19 (1999), 255-266.
%D A000125 D. J. Price, Some unusual series occurring in n-dimensional geometry, 
               Math. Gaz., 30 (1946), 149-150.
%D A000125 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000125 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000125 T. H. Stickels, Mindstretching Puzzles. Sterling, NY, 1994 p. 85.
%D A000125 W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, 
               p. 30.
%D A000125 A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with 
               Elementary Solutions. Vol. I. Combinatorial Analysis and Probability 
               Theory. New York: Dover Publications, Inc., 1987, p. 13, #45 (First 
               published: San Francisco: Holden-Day, Inc., 1964)
%H A000125 T. D. Noe, <a href="b000125.txt">Table of n, a(n) for n=0..1000</a>
%H A000125 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000125 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000125 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000125 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some 
               Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), 
               Article 02.1.7
%H A000125 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CylinderCutting.html">Link to a section of The World of Mathematics 
               (1).</a>
%H A000125 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SpaceDivisionbyPlanes.html">Link to a section of The World of Mathematics 
               (2).</a>
%H A000125 R. Zumkeller, <a href="a161700.txt">Enumerations of Divisors</a> [From 
               Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]
%F A000125 (n+1)*(n^2-n+6)/6 = (n^3 + 5n + 6) / 6.
%F A000125 G.f.: (1-2x+2x^2)/(1-x)^4; - Paul Barry (pbarry(AT)wit.ie), Jun 21 2005
%F A000125 C(n,3)+C(n,1). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 24 
               2006
%F A000125 a(n) = sum of (n+1)-th row terms of A077028. Also, binomial transform 
               of [1,1,1,1,0,0,0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 23 2007
%F A000125 a(n)=C(n+3,n)+n,n>=1 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 24 2009]
%e A000125 a(4)=15 because there are 15 compositions of 5 into four or fewer parts. 
               a(6)=42 because the sum of the first four terms in the 6th row of 
               Pascal's triangle is 1+6+15+20=42 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Jan 23 2009]
%p A000125 A000125 := n->(n+1)*(n^2-n+6)/6;
%p A000125 seq(binomial(n,3)+binomial(n,2)+binomial(n,1)+binomial(n,0), n=0..29); 
               - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
%p A000125 A000125:=(1-2*z+2*z**2)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%t A000125 a=1; lst={a}; k=1; e=0; Do[a+=k; AppendTo[lst, a]; e++; k+=e, {n, 0, 
               6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 
               2008]
%Y A000125 Cf. A000124, A003600. Bisections give A100503, A100504.
%Y A000125 Cf. A077028.
%Y A000125 A005408, A000124, A016813, A086514, A058331, A002522, A161701, A161702, 
               A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, 
               A161711, A161712, A161713, A161715, A006261. [From Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]
%Y A000125 Sequence in context: A082562 A159243 A089140 this_sequence A129961 A133551 
               A114226
%Y A000125 Adjacent sequences: A000122 A000123 A000124 this_sequence A000126 A000127 
               A000128
%K A000125 nonn,easy,nice
%O A000125 0,2
%A A000125 N. J. A. Sloane (njas(AT)research.att.com).
%E A000125 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 22 2000